On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 23 of 188
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To obtain the unit outward normal n_ along a line a3, a transformation matrix
To = cos O - sin ] (1.20)
sino cos (
By multipling To to any vector v, a new vector vo rotated 0 degree with respect
to v is obtained. Let a# = (xo - x,, yo - y,) represent the vector from node a to
node #3; then
na = TO _(1.21)
is the unit vector formed by rotating a by 0 degrees. To find the unit outward
normal n of the control volume b,,m along cij, we let 0 = -r/2 then,
T~ cj _ (Y Yi) i+(xij -x )
2 cij ci cij -
where ca] = (xC - xU, yc - y,,). Therefore,
fz = Pup Aup [- (kxx + kxy o
clij . (1.22)
- ky x+ kyy (-] ds.
The phase potential in (1.22) is approximated by h E P'(T) and
h(1) = Li(L) o + Lj (1) o + Lk(1) k
where oj, oj, and pk are the phase potential values at triangular vertices. Con-
sequentlly, the derivatives of p, are constants in t E T; therefore, (1.22) can be
f, = pupAup - kx + kxy (Yc - yia)
- (kyOx + kyy+) (xzi - xc)] .
The flux f2 can be derived in the same way as f2 and as a result f2 is fully
defined. Let S be approximated by Sh E P0(B) then Sh(b2) = Si where Si is the
saturation value of control volume b2. The partial residual function of bim can
then be written as
fibim( pS2 (1.24)
F = f + b2,m ,.24)t
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Deo, Milind D. On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs, report, August 31, 2005; Utah. (https://digital.library.unt.edu/ark:/67531/metadc877059/m1/23/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.